Exponent Calculator

Exponent Calculator 

With this exponent calculator, you can find the result of both positive and negative exponents. The input system is simple and easy to understand. By clicking on “show more”, you can also see all the steps of the calculation.

What is the exponent?

Exponents can be defined as,

“The exponent of a number says how many times to use the number in a multiplication.”

For example,  3 exp 4 can be written as:

3⁴ = 3×3×3×3 = 81

How to solve exponents?

There are two types of exponents; positive and negative. 

Positive exponents

Exponents of a base that are equal to or greater than zero.

Example:

Solve 5³.

Solution:

Step 1: Write the base and multiply it up to power times.

5³ = 5 × 5 × 5 (5 to the 3rd power)

Step 2: Multiply.

5³ = 5 × 5 × 5 = 125 

Example

Solve 2^23624

Solution:

Step 1: Write 2 up to 23624 times with a multiply sign between each term of 2.

 2^23624 = 2 x 2 x 2 x 2 x 2 x ... x 2 (23624 times)

Step 2: Multiply.

 2^23624 = infinity (too large number)

Example

1.1 raised to 10

Solution:

1.1 raised to 10 = 2.5937

Negative exponents

Exponents that are smaller than zero.

Example:

Solve 6^-3

Solution:

Step 1: Divide 6^-3 by 1 to make the exponent positive.

6^-3 = 1/6³ (6 to the 3rd power)

Step 2: Write the base and multiply it up to power times.

1/6³ = 1/(6 × 6 × 6) = 1/216 

1/6³ = 0.00463

This is the basics of exponents. On digging a little deeper, you get the rules or laws of the exponents. These rules help to minimize the steps required for simplification. 

If you do not use these rules and solve the exponents on your primary knowledge of PEMDAS, you will still get the same result but just a little later.

Laws of Exponents:

Some say there are 7 laws while some believe there are 8 laws. There are also the ones who write 10 laws of exponents. 

In the end, it is just the number of shortcuts you know to accelerate the calculations. The count does not really matter. 

For your ease, we have divided the exponent rules into categories. i.e. Simple and complicated.

Simple rules:

You can call them the introductory or common sense laws too ????. These three rules are:

• X¹ = X (Which makes sense because there is only one x).
• X⁰ = 1 (No x, no multiplication. Let’s say it’s just a fancy way to write one).
• X^-1 = 1/X (This rule has been covered in negative exponents already).

Disclaimer: This Category and the name of the rules are not mentioned by any mathematician. These are just devised to make the rules understandable.

Complicated rules

There are six main rules that fall under this heading. They may look difficult but by paying some attention, you can understand them alright. 

Rule no 1:

(b^m)(bⁿ) = b^m+n

This rule also goes by the name “Product law”. When the bases are the same, different powers are added. 

It can be explained by thinking about how many times the b needs to be multiplied. Let’s take an example. 

= (Y³)(Y²)

The Y is first multiplied three times and then 2 times. 

= (Y x Y x Y)(Y x Y)

This means the final answer will be equal both ways. It is because the value Y in a particular problem represents a specific digit. It is not possible to say Y is equal to 5 and then Y is also equal to 10.

Since they have the same values, the powers are combined.
 Unless the problem in question is related to roots.  

Rule no 2:

(b^m)/(bⁿ) = b^m-n

Very similar to the previous law, this law is called “Quotient Law”. For its explanation, take the previous example with a division sign. 

= (Y³)/(Y²)
= (Y x Y x Y)/(Y x Y)

The two Ys in the upper value will be canceled out by the Ys in the lower value. That leaves only one Y. Same way when 2 is subtracted from 3, you get one Y. 

Rule no 3:

(axb)^m = (a)^m(b)^m

“Power of a product rule”. When two values in a product have the same power, they can be separated. 

Whether you multiply first and then solve the power or solve the powers separately and then multiply, the result will be common. 

Rule no 4:

(a/b)^m = (a)^m/(b)^m

Hopefully, it makes sense after the previous law. Its name is the “Power of a quotient rule”. 

Rule no 5:

((b)^m)ⁿ = b^mxn 

The “Power to a power rule” states that when the base(b) is raised to two powers, first m then n, the powers are multiplied. It is a little hard to comprehend it but you can see the verification example below.

=  (2²)³ = 2^2x3 = 26 = 64

Now, without implementing the law.  

= (22)³ = (4)³ = 64

Rule no 6:

b^m/n = ⁿ√b^m 

This is the “fractional power rule”. When the power is in the form of a fraction, it is nearly impossible to solve it by hand. This is why it is changed into the square root form.

As you may know, Y^1/n = ⁿ√Y. This justifies the rule if b^m is considered Y.

That is it, the laws of the exponents.

How to evaluate the nth power of a number?

Let's say we have the nth power of base value is "b." Then for nth power b is multiplies to itself for n time. Suppose we have to find the 4rth power of b, then it will be b*b*b*b*b.

Some of the examples solved by this nth power calculator are given in the below table.